Theorem conncompss 23357

Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompss	⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1133	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑋)
2		conntop 23341	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Conn → (𝐽 ↾t 𝑇) ∈ Top)
3	2	3ad2ant3 1132	. . . . . 6 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐽 ↾t 𝑇) ∈ Top)
4		restrcl 23081	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
5	4	simprd 494	. . . . . 6 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → 𝑇 ∈ V)
6		elpwg 4609	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
7	3, 5, 6	3syl 18	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
8	1, 7	mpbird 256	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9		3simpc 1147	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn))
10		eleq2 2818	. . . . . 6 ⊢ (𝑦 = 𝑇 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑇))
11		oveq2 7434	. . . . . . 7 ⊢ (𝑦 = 𝑇 → (𝐽 ↾t 𝑦) = (𝐽 ↾t 𝑇))
12	11	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = 𝑇 → ((𝐽 ↾t 𝑦) ∈ Conn ↔ (𝐽 ↾t 𝑇) ∈ Conn))
13	10, 12	anbi12d 630	. . . . 5 ⊢ (𝑦 = 𝑇 → ((𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn) ↔ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
14		eleq2 2818	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
15		oveq2 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
16	15	eleq1d 2814	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
17	14, 16	anbi12d 630	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
18	17	cbvrabv 3441	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)}
19	13, 18	elrab2 3687	. . . 4 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
20	8, 9, 19	sylanbrc 581	. . 3 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
21		elssuni 4944	. . 3 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
22	20, 21	syl 17	. 2 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
23		conncomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
24	22, 23	sseqtrrdi 4033	1 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   ⊆ wss 3949  𝒫 cpw 4606  ∪ cuni 4912  (class class class)co 7426   ↾_t crest 17409  Topctop 22815  Conncconn 23335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-rest 17411  df-top 22816  df-conn 23336

This theorem is referenced by:  conncompcld  23358  tgpconncompeqg  24036  tgpconncomp  24037